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In the realm of complex numbers, expressing a number in its standard form is crucial for simplification and understanding. The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) denotes the imaginary unit. This form

• clearly distinguishes between the real part, \(a\), and the imaginary part, \(bi\).
• enables straightforward addition, subtraction, and comparison of complex numbers.

To convert a product of complex numbers or expressions into standard form, you rearrange the result to separate the components. For instance, after performing operations, you ensure that terms involving \(i\) are presented clearly as the second component. In our given example \(-40 - 35i\), \(a = -40\) is the real part, and \(b = -35\) is the imaginary coefficient. By expressing results in standard form, calculation and comprehension become more manageable.

Distribution is a fundamental concept in algebra that simplifies expressions involving parentheses. The distributive property states that \(a(b + c) = ab + ac\). When dealing with complex numbers, distribution allows us to effectively multiply terms. This process involves

• expanding expressions by multiplying each term inside the parentheses by a term outside.
• reducing compound expressions into simpler parts.

In the scenario of \(-5i(7 - 8i)\), distribution means multiplying \(-5i\) by each term in the expression \(7 - 8i\). It's crucial to handle each part separately, including correctly applying rules involving imaginary units. Proper distribution results in expressions like \(-35i\) and \(40i^2\). Understanding distribution is key to managing complex equations efficiently, particularly when they involved imaginary numbers.

Multiplying complex numbers involves a few straightforward steps but requires attention to detail. In our example, we multiply \(-5i\) by \(7 - 8i\). The steps are straightforward:

• Multiply the real and imaginary components separately.
• Remember that \(i^2 = -1\), an essential identity in complex arithmetic.

First, we multiply \(-5i\) by \(7\) to get \(-35i\). Then, \(-5i\) times \(-8i\) gives \(40i^2\), which becomes \(-40\) since \(i^2 = -1\). The key is combining these products to form a complete expression. Through these steps, we achieve a result of \(-40 - 35i\). Multiplying complex numbers requires an understanding of how real and imaginary parts interact, especially when the imaginary unit \(i\) squares to a negative number. This method ensures you correctly integrate each part of the numbers involved, leading to a reliable result in the standard format.